Phase locked loops are common in many systems such as, for example, receivers in communication systems. Phase locked loops can be used to provide a local oscillator signal which is locked in frequency and phase to a carrier signal of a received communications signal. If the received signal comprises an information signal multiplied by a carrier signal, the received signal is typically mixed with (multiplied by) the local oscillator signal and low-pass filtered to recover the information signal.
A QPSK (Quadrature Phase Shift Keying) modulated signal typically comprises a signal of constant frequency which can have one of four phases, each phase representing two bits. It can be generated, for example, by multiplying two binary information signals, an in-phase signal I(t) and a quadrature-phase signal Q(t), by a carrier signal at a phase of 0° and 90° respectively (i.e. a cosine signal and a sine signal respectively). The resulting signals are then summed to form the QPSK signal.
An ideal received QPSK signal typically comprises the signal R(t) where:R(t)=I(t)·cos(ω1t+φ1)+Q(t)·sin(ω1t+φ1)  (1)where:                I(t) is the transmitted in-phase signal,        Q(t) is the transmitted quadrature-phase signal,        ω1 is the carrier signal frequency, and        φ1 is the received carrier static phase.        
FIG. 1 shows a typical Costas phase locked loop 100 which can be used to demodulate a QPSK signal. The Costas Loop 100 comprises a first mixer 102 which mixes a received signal R(t) with a local oscillator signal provided by a voltage controlled oscillator (VCO) 104. The Costas Loop 100 also comprises a second mixer 106 which mixes the received signal R(t) with the local oscillator signal delayed by 90°, as represented by a 90° delay unit 108.
The output of the first mixer is low-pass filtered by a first low-pass filter (LPF) 110. The output of the second mixer is low-pass filtered by a second low-pass filter 112. If the VCO 104 is generating a sinusoidal signal which is locked in frequency and phase to the incoming signal R(t) (i.e. it is generating a signal proportional to cos(ω1t+φ1)), then the output of the first LPF 110 will be the signal I(t), and the output of the second LPF will be the signal Q(t).
To achieve frequency and phase lock, the outputs of the filters 110 and 112 are provided to a phase detection apparatus 114 which is, in practice, a third mixer. The output of the third mixer is provided to a loop filter 116 which is, in practice, a third low pass filter. The output of the filter 116 is a control voltage which is applied to the VCO 104 to control the frequency of its output. Over a period of time, the frequency and phase of the local oscillator signal provided by the VCO 104 will approach those of the carrier signal of the received signal R(t). The receiver is then said to be carrier synchronized with the transmitter.
A typical QPSK demodulator also must achieve timing synchronization (also known as symbol synchronization). The signals Q(t) and I(t) must be sampled at periodic instants to determine their amplitude and therefore the information that the signals are representing. Timing synchronization is typically performed separately from carrier synchronization in QPSK demodulation.
OQPSK (Offset QPSK) is another modulation scheme where the signal Q(t) or I(t) is delayed by half a symbol when the OQPSK is modulated. In the subsequent transmitted symbol, phase shifts are restricted to a maximum of 90° at a time. This gives OQPSK desirable properties over QPSK which can have phase shifts of 180°.
Some OQPSK synchronization schemes introduce a half-symbol delay to the signal I(t) in the carrier synchronization loop. However, the half-symbol delay introduces a carrier phase shift which is no longer negligible when a large carrier frequency offset (difference in frequency between the local oscillator signal and the carrier signal of a received signal R(t)) is present. Therefore these schemes are unsuitable where a large carrier frequency offset is present.
OQPSK synchronization schemes are described in Peter W. Kinman, Jeff B. Berner, “Carrier Synchronization of Offset QPSK for Deep Space Telemetry”, Aerospace Conference Proceedings, 2002. IEEE, Volume 3, 9-16, March 2002, pages 3-1327 to 3-1336, and M. K. Simon, “Carrier Synchronization of Offset Quadrature Phase Shift Keying”, Telecommunications and Mission Operations Progress Report 42-133, January-March 1998, May 15, 1998. http://tmo.jpl.nasa.gov/tmo/progress_report/42-133/133J.pdf